Local semiconvexity of Kantorovich potentials on non-compact manifolds

Alessio Figalli; Nicola Gigli

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 3, page 648-653
  • ISSN: 1292-8119

Abstract

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We prove that any Kantorovich potential for the cost function c = d2/2 on a Riemannian manifold (M, g) is locally semiconvex in the “region of interest”, without any compactness assumption on M, nor any assumption on its curvature. Such a region of interest is of full μ-measure as soon as the starting measure μ does not charge n – 1-dimensional rectifiable sets.

How to cite

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Figalli, Alessio, and Gigli, Nicola. "Local semiconvexity of Kantorovich potentials on non-compact manifolds." ESAIM: Control, Optimisation and Calculus of Variations 17.3 (2011): 648-653. <http://eudml.org/doc/272798>.

@article{Figalli2011,
abstract = {We prove that any Kantorovich potential for the cost function c = d2/2 on a Riemannian manifold (M, g) is locally semiconvex in the “region of interest”, without any compactness assumption on M, nor any assumption on its curvature. Such a region of interest is of full μ-measure as soon as the starting measure μ does not charge n – 1-dimensional rectifiable sets.},
author = {Figalli, Alessio, Gigli, Nicola},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Kantorovich potential; optimal transport; regularity},
language = {eng},
number = {3},
pages = {648-653},
publisher = {EDP-Sciences},
title = {Local semiconvexity of Kantorovich potentials on non-compact manifolds},
url = {http://eudml.org/doc/272798},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Figalli, Alessio
AU - Gigli, Nicola
TI - Local semiconvexity of Kantorovich potentials on non-compact manifolds
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2011
PB - EDP-Sciences
VL - 17
IS - 3
SP - 648
EP - 653
AB - We prove that any Kantorovich potential for the cost function c = d2/2 on a Riemannian manifold (M, g) is locally semiconvex in the “region of interest”, without any compactness assumption on M, nor any assumption on its curvature. Such a region of interest is of full μ-measure as soon as the starting measure μ does not charge n – 1-dimensional rectifiable sets.
LA - eng
KW - Kantorovich potential; optimal transport; regularity
UR - http://eudml.org/doc/272798
ER -

References

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  1. [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000). Zbl0957.49001MR1857292
  2. [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in spaces of probability measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2005). Zbl1090.35002MR2129498
  3. [3] D. Cordero-Erasquin, R.J. McCann and M. Schmuckenschlager, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math.146 (2001) 219–257. Zbl1026.58018MR1865396
  4. [4] A. Fathi and A. Figalli, Optimal transportation on non-compact manifolds. Israel J. Math. (to appear). Zbl1198.49044MR2607536
  5. [5] A. Figalli, Existence, uniqueness, and regularity of optimal transport maps. SIAM J. Math. Anal.39 (2007) 126–137. Zbl1132.28322MR2318378
  6. [6] W. Gangbo and R.J. McCann, The geometry of optimal transportation. Acta Math.177 (1996) 113–161. Zbl0887.49017MR1440931
  7. [7] N. Gigli, Second order analysis on ( 𝒫 2 ( M ) , W 2 ) . Memoirs of the AMS (to appear), available at http://cvgmt.sns.it/cgi/get.cgi/papers/gig09/. Zbl1253.58008
  8. [8] R.J. McCann, Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal.11 (2001) 589–608. Zbl1011.58009MR1844080
  9. [9] C. Villani, Optimal transport, old and new, Grundlehren des mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer-Verlag, Berlin-New York (2009). Zbl1156.53003MR2459454

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